\documentclass[reqno]{amsart}
\usepackage{hyperref}


\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 132, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/132\hfil Strong resonance problems]
{A note on strong resonance problems for p-Laplacian}

\author[C. H. Jin, Y. Y. Ke, J. X. Yin\hfil EJDE-2006/132\hfilneg]
{Chunhua Jin, Yuanyuan Ke, Jingxue Yin}

\address{Chunhua Jin \newline
Department of Applied Mathematics, Jilin University,
Changchun 130012, China}
\email{diffusion@jlu.edu.cn}

\address{Yuanyuan Ke \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou 510275, China. \newline
Department of Applied Mathematics, Jilin University,
Changchun 130012, China}
\email{keyy@jlu.edu.cn (corresponding author)}

\address{Jingxue Yin\newline
Department of Applied Mathematics, Jilin University,
Changchun 130012, China}
\email{yjx@jlu.edu.cn}


\date{}
\thanks{Submitted March 21, 2006. Published October 17, 2006.}
\thanks{Supported by the NSFC, NSFGD-06300481, China Postdoctoral Science Foundation, and 
 \hfill\break\indent
 the Specific Foundation for Ph.D. Specialities of Educational Department of China}
\subjclass[2000]{35G30, 35A15, 35B38}
\keywords{p-Laplacian equations; boundary value problem; eigenvalue;
\hfill\break\indent strong resonance problems}

\begin{abstract}
 In this note, we study the existence of the weak solutions for the
 $p$-Laplacian with strong resonance, which generalizes the 
 previous results in one-dimension.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In a previous paper, Bouchala \cite{1} studied the existence of the weak
solutions of the nonlinear boundary-value problem for one-dimensional case
\begin{gather*}
-\Delta_pu=\lambda |u|^{p-2}u+g(u)-h(x), \quad x\in (0,\pi),\\
u(0)=u(\pi)=0,
\end{gather*}
where $p>1$, $\lambda\in\mathbb{R}$, $h\in L^{p'}(0, \pi)$ ($p'=\frac p{p-1}$),
and $g: \mathbb{R}\to \mathbb{R}$ is a continuous and nonlinear function of
the Landesman-Lazer type.
By applying the variational approach, the author translated
problem into a critical points problem, and proved the existence of critical points
separately for situations
$$
\lambda<\lambda_1, \quad \lambda_k<\lambda<\lambda_{k+1},
\quad \lambda=\lambda_k,
$$
where $\{\lambda_k\}$ is the sequence of eigenvalues and satisfies
$0<\lambda_k<\lambda_{k+1}$. The results extended a previous result by
J.~Bouchala and P.~Dr$\acute{\rm a}$bek \cite{5}, in which, they only considered
the case of $\lambda=\lambda_1$, that is, $\lambda$ is the first eigenvalue.

The researches on the existence of weak solutions for the resonance problem
to $p$-Laplacian can also be found in the other papers, such as
\cite{2,3}  and the references therein.
In \cite{2}, which examined resonance
problems at arbitrary eigenvalues for the analogous ODE problem. However,
in \cite{3},
the author not only generalized the results in \cite{2} into
higher-dimension, but also proved the existence of weak solutions for
the case of $\lambda\in \mathbb{R}$, that is $\lambda$ is not only an eigenvalue.

In this short note, we would like to point a fact that the existence
results that J.~Bouchala has proved in \cite{1}
are also true for the higher dimensional case. In fact,
by substituting the higher dimensional domain $\Omega$ for the one-dimensional
interval $(0, \pi)$, we may consider the following boundary-value problem
\begin{equation} \label{1-1}
\begin{gathered}
-\Delta_pu=\lambda |u|^{p-2}u+g(u)-h(x), \quad x\in \Omega, \\
u|_{\partial \Omega}=0,
\end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ is a bounded domain with smooth boundary,
$\lambda \in\mathbb{R}$, $N\ge 1$, $p>1$, $g:\mathbb{R}\to\mathbb{R}$
is a continuous function, $h\in L^{p'}(\Omega)$ ($p'=\frac p{p-1}$), and
$\Delta_p$ is the $p$-Laplacian operator, that is
$\Delta_p u =\mathop{\rm div} (|\nabla u|^{p-2}\nabla u)$.
Similar to \cite{1}, we say that $\lambda\in \mathbb{R}$ is an eigenvalue of
$-\Delta_p$, if there exists a nonzero function $u\in W_0^{1, p}(\Omega)$, such
that
$$
\int_\Omega |\nabla u|^{p-2}\nabla u \nabla v \,dx
=\lambda\int_\Omega|u|^{p-2}uv\,dx \quad \text{for all } v\in W_0^{1,p}(\Omega).
$$
The function $u$ is called an eigenfunction of $-\Delta_p$ corresponding to
the eigenvalue  $\lambda$, and we denote it by
$$
u\in \ker (-\Delta_p-\lambda)\backslash\{0\}.
$$

For convenience, we first introduce some notation.
Consider the functional $R: W_0^{1,p}(\Omega)\backslash \{0\}\to \mathbb{R}$,
$$
R(u)=\frac{\int_\Omega|\nabla u|^p \,dx}{\int_\Omega |u|^p\,dx},
\quad u\in W_0^{1,p}(\Omega)\backslash\{0\},
$$
and the manifold
$$
\mathcal{S}=\{u\in W_0^{1,p}(\Omega): \|u\|_{L^p(\Omega)}=1\}.
$$
For $k\in \mathbb N$, let
$$
\mathcal F_k:=\{\mathcal A\subset \mathcal{S}:  \text{there exists a continuous
odd surjection} \ h: \mathcal{S}_{k-1}\to \mathcal A\},
$$
where $\mathcal{S}_{k-1}$ represents the unit sphere in $\mathbb{R}^{k}$.
Let
$$
\lambda_k=\inf_{\mathcal A\in
\mathcal F_k}\sup_{u\in \mathcal A} R(u).
$$
It is known that $\lambda_k$ is an eigenvalue of $-\Delta_p$, and
$0<\lambda_k<\lambda_{k+1}$ (see \cite{3, 4, 6}).
Here, we denote the norm in $W_0^{1,p}(\Omega)$ by
$$
\|u\|=\Big(\int_\Omega |\nabla u|^p \,dx\Big)^{1/p} \quad
\text{for all } u\in W_0^{1,p}(\Omega).
$$
By Poincar\'{e}'s inequality, we see that the norm $\|\cdot\|$ parallels to the
usual definition. Furthermore, we denote
\begin{equation} \label{1-2}
F(u)=\begin{cases}
\frac pu\int_0^ug(s)ds-g(u), & u\not=0, \\
(p-1)g(0), & u=0,
\end{cases}
\end{equation}
and set
\begin{gather*}
\overline {F(-\infty)}=\limsup_{u\to -\infty}F(u), \quad
\underline {F(-\infty)}=\liminf_{u\to -\infty}F(u), \\
\overline {F(+\infty)}=\limsup_{u\to +\infty}F(u), \quad
\underline {F(+\infty)}=\liminf_{u\to +\infty}F(u).
\end{gather*}
Throughout this paper, we assume:
(i)
\begin{equation} \label{1-3}
\lim_{|t|\to \infty}\frac{g(t)}{|t|^{p-1}}=0.
\end{equation}
(ii) For any $v\in \ker (-\Delta_p-\lambda)\backslash\{0\}$,
%%
\begin{equation}\label{1-4}
(p-1)\int_{\Omega}h(x)v(x)\,dx<\underline {F(+\infty)}\int_{\Omega}v^+(x)\,dx
+\overline {F(-\infty)}\int_{\Omega}v^-(x)\,dx,
\end{equation}
or for every $v\in \ker (-\Delta_p-\lambda)\backslash\{0\}$,
\begin{equation} \label{1-5}
(p-1)\int_{\Omega}h(x)v(x)\,dx>\overline {F(+\infty)}\int_{\Omega}v^+(x)\,dx
+\underline {F(-\infty)}\int_{\Omega}v^-(x)\,dx,
\end{equation}
where $v^+=\max\{0, v\}$, $v^-=\min\{0, v\}$.

The following theorem is the main result of this note.

\begin{theorem}\label{1.1}
If \eqref{1-3}, \eqref{1-4} (or \eqref{1-5}) hold, then problem
\eqref{1-1} admits at least one weak solution.
\end{theorem}

\begin{remark}
If $\lambda$ is not an eigenvalue of $-\Delta_p$, then \eqref{1-4}, \eqref{1-5}
are vacuously true.
\end{remark}

\section{Proof of Main Result}

To employ the variational approach, we introduce the functional
$$
J_\lambda(u):=\frac1p\int_\Omega|\nabla u|^p\,dx-\frac\lambda p\int_\Omega|u|^p\,dx
-\int_\Omega G(u)\,dx+\int_\Omega h(x)u(x)\,dx,
$$
where $G(t)=\int_0^tg(s)ds$.
Clearly, $J_\lambda\in C_1(W_0^{1,p}(\Omega); \mathbb{R})$, and for every
$v\in W_0^{1,p}(\Omega)$,
$$
\langle J_\lambda'(u), v\rangle=\int_\Omega|\nabla u|^{p-2}\nabla u\nabla v\,dx
-\lambda\int_\Omega |u|^{p-2}uv \,dx
-\int_{\Omega}g(u)v\,dx +\int_\Omega hv\,dx.
$$
Note that the weak solutions of \eqref{1-1} correspond to the
critical points of $J_\lambda$.

To show that $J_\lambda$ has critical points of saddle point type,
we need a fundamental lemma as follows. (see \cite{3} or \cite{7})

\begin{lemma}[Deformation Lemma]
Suppose that $J_\lambda$ satisfies the Palais-Smale condition,
i.e. if $\{u_n\}$ is a sequence of functions in $W_0^{1,p}(\Omega)$
such that $\{J_\lambda(u_n)\}$ is bounded in $\mathbb{R}$,
 and $J'_\lambda(u_n)\to 0$ in $(W_0^{1,p}(\Omega))^*$,
then $\{u_n\}$ has a subsequence that is strongly convergent in
 $W_0^{1,p}(\Omega)$. Let $c\in\mathbb{R}$ be a regular value of
$J_\lambda$ and let $\bar \varepsilon>0$. Then there exists
$\varepsilon\in (0, \bar\varepsilon)$ and a
continuous one-parameter family of homeomorphisms,
$\phi: W_0^{1,p}(\Omega)\times [0, 1]\to W_0^{1,p}(\Omega)$
with the properties:
\begin{itemize}
\item[(i)] If $t=0$ or if $|J_\lambda(u)-c|\ge \bar\varepsilon$, then
$\phi(u,t)=u$;

\item[(ii)] if $J_\lambda(u)\le c+\varepsilon$, then
$J_\lambda(\phi(u,1))\le c-\varepsilon$.
\end{itemize}
\end{lemma}

The following lemma is a crucial step of our argument.
%%
\begin{lemma}
\label{Lem-1}
Assume \eqref{1-3} and \eqref{1-4} (or \eqref{1-5}) hold. Then the functional
$J_\lambda$ satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof}
Assume that $\{u_n\}$ is a sequence of functions in $W^{1,p}_0(\Omega)$,
and there exists an positive constant $M$ such that
\begin{gather} \label{3.1}
|J_\lambda(u_n)|\le M, \\
\label{3.2}
J_\lambda'(u_n)\to  0\qquad \hbox{ in }(W^{1,p}_0(\Omega))^*.
\end{gather}
%%
In the following, we shall show that the Palais-Smale sequence $\{u_n\}$
is bounded.
Suppose to the contrary (passing to the subsequence if necessary), namely
$$
\|u_n\|\to +\infty.
$$
Let $v_n:=\frac{u_n}{\|u_n\|}$.
Due to the reflexivity of $W^{1,p}_0(\Omega)$ and the compact embedding
$$
W^{1,p}_0(\Omega)\hookrightarrow L^p(\Omega),
$$
there exists $v\in W^{1,p}_0(\Omega)$ such that (passing to subsequences)
%%
\begin{gather} \label{3.4}
v_n\rightharpoonup v\quad \hbox{in } W^{1,p}_0(\Omega), \\
\label{3.5}
v_n\to  v\quad\hbox{in } L^p(\Omega).
\end{gather}
%%
 From (\ref{3.2}) and (\ref{3.4}), we have
\begin{equation} \label{3.6}
\begin{aligned}
0&\leftarrow \frac{\langle J'_\lambda(u_n),v_n-v\rangle}{\|u_n\|^{p-1}}
\\
=&\int_{\Omega}|\nabla v_n|^{p-2}\nabla v_n(\nabla v_n-\nabla v)\,dx
-\lambda\int_{\Omega}|v_n|^{p-2}v_n(v_n-v)\,dx \\
&\quad -\int_{\Omega}\frac{g(u_n)}{\|u_n\|^{p-1}}(v_n-v)\,dx
+\int_{\Omega}\frac{h}{\|u_n\|^{p-1}}(v_n-v)\,dx.
\end{aligned}
\end{equation}
Since \eqref{1-3} and (\ref{3.5}), it follows that
the last three terms approach to $0$ as $n\to\infty$. Then we have
$$
\int_{\Omega}|\nabla v_n|^{p-2}\nabla v_n(\nabla v_n-\nabla v)\,dx\to 0.
$$
Furthermore, we have
%%
\begin{equation}
\label{3.7}
\begin{aligned}
0&\leftarrow  \int_{\Omega}|\nabla v_n|^{p-2}\nabla v_n(\nabla v_n-\nabla v)\,dx
-\int_{\Omega}|\nabla v|^{p-2}\nabla v(\nabla v_n-\nabla v)\,dx \\
&= \int_\Omega|\nabla v_n|^p\,dx-\int_\Omega|\nabla v_n|^{p-2}
  \nabla v_n\nabla v\,dx
-\int_\Omega|\nabla v|^{p-2}\nabla v\nabla v_n\,dx+\int_\Omega|\nabla v|^p\,dx\\
&\ge  \|v_n\|^p-\|v_n\|^{p-1}\|v\|-\|v\|^{p-1}\|v_n\|+\|v\|^p
\\
&= (\|v_n\|^{p-1}-\|v\|^{p-1})(\|v_n\|-\|v\|)\ge0,
\end{aligned}
\end{equation}
which implies
%%
\begin{equation}
\label{3.8}
\|v_n\|\to \|v\|, \quad n\to\infty.
\end{equation}
%%
Noticing that $v_n\rightharpoonup v$ in $W^{1,p}_0(\Omega)$, and combining with
the uniform convexity of $W^{1,p}_0(\Omega)$, we infer that
%%
\begin{equation}
\label{3.9}
v_n\to  v\quad \hbox{ in }W^{1,p}_0(\Omega), \quad\|v\|=1.
\end{equation}
%%
Moreover, for any $w\in W^{1,p}_0(\Omega)$, as $n\to\infty$,
%%
\begin{align*}
\frac{\langle J'_\lambda(u_n),w\rangle}{\|u_n\|^{p-1}}
&=\int_\Omega|\nabla v_n|^{p-2}\nabla v_n\nabla w\,dx
-\lambda\int_\Omega|v_n|^{p-2}v_nw\,dx\\
&\quad -\int_\Omega\frac{g(u_n)}{\|u_n\|^{p-1}}w\,dx
+\int_\Omega\frac{h}{\|u_n\|^{p-1}}w\,dx\to 0.
\end{align*}
%%
Clearly the last two terms approach to zero. Hence for all $w\in W^{1,p}_0(\Omega)$:
%%
\begin{equation}\label{3.10}
\int_\Omega|\nabla v_n|^{p-2}\nabla v_n\nabla w\,dx-
\lambda\int_\Omega|v_n|^{p-2}v_nw\,dx\to  0, \quad\text{as } n\to\infty,
\end{equation}
%%
which implies
$$
\int_\Omega|\nabla v|^{p-2}\nabla v\nabla w\,dx=\lambda\int_\Omega|v|^{p-2}vw\,dx,
\quad\forall \ w\in W^{1,p}_0(\Omega)
$$
and
$v\in \ker (-\Delta_p-\lambda)\backslash\{0\}$, $\|v\|=1$.
The boundedness of $\{J_\lambda(u_n)\}$, $J'_\lambda(u_n)\to 0$, and
$\|u_n\|\to \infty$ imply
%%
\begin{align*}
0&\leftarrow  \frac{\langle J'_\lambda(u_n), u_n\rangle
-pJ_\lambda(u_n)}{\|u_n\|}\\
&= \int_\Omega\frac{pG(u_n)-g(u_n)u_n}{\|u_n\|}\,dx
-(p-1)\int_\Omega h\frac{u_n}{\|u_n\|}\,dx\\
&= \int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx
-(p-1)\int_\Omega h\frac{u_n}{\|u_n\|}\,dx,
\end{align*}
%%
that is,
%%
\begin{equation}
\label{3.11}
\lim_{n\to\infty}\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx=(p-1)
\int_\Omega hv\,dx.
\end{equation}
%%
Now we assume that \eqref{1-4} (the other case \eqref{1-5} can be treated similarly)
holds. It follows that
$$
\underline{F(+\infty)}>-\infty \quad\hbox{and}\quad
\overline{F(-\infty)}<+\infty.
$$
For arbitrary $\varepsilon>0$, set
\begin{gather*}
c_\varepsilon:=\begin{cases}
 \underline{F(+\infty)}-\varepsilon &\hbox{if }
 \underline{F(+\infty)}\in\mathbb{R}, \\
1/\varepsilon &\hbox{if }\underline{F(+\infty)}=+\infty;
\end{cases} \\
d_\varepsilon:=\begin{cases}
 \overline{F(-\infty)}+\varepsilon &\hbox{if }
 \overline{F(-\infty)}\in\mathbb{R}, \\
 -1/\varepsilon &\hbox{if }\overline{F(-\infty)}=-\infty.
\end{cases}
\end{gather*}
Then for every $\varepsilon>0$ there exists $K>0$ such that
\begin{equation}\label{3.12}
\begin{gathered}
F(t)\ge c_\varepsilon \quad\hbox{ for all }  t>K,\\
F(t)\le d_\varepsilon \quad\hbox{ for all }  t<-K.
\end{gathered}
\end{equation}
%%
On the other hand, the continuity of $F$ on $\mathbb{R}$ implies
that for any $K>0$ there exists $c(K)>0$ such that
%%
\begin{equation} \label{3.13}
|F(t)|\le c(K)\quad \hbox{for all } t\in[-K, K].
\end{equation}
%%
Choose $\varepsilon>0$ and consider the corresponding $K>0$ and $c(K)>0$
given by (\ref{3.12}) and (\ref{3.13}), respectively. Set
%%
\begin{equation}
\label{3.14}
\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx
=A_{K,n}+B_{K,n}+C_{K,n}+D_{K,n}+E_{K,n},
\end{equation}
where
\begin{gather*}
A_{K,n}=\int_{\{x\in\Omega:|u_n(x)|\le K\}}F(u_n)\frac{u_n}{\|u_n\|}\,dx,\\
B_{K,n}=\int_{\{x\in\Omega:u_n(x)>K,v(x)>0\}}F(u_n)\frac{u_n}{\|u_n\|}\,dx, \\
C_{K,n}=\int_{\{x\in\Omega:u_n(x)>K,v(x)\le0\}}F(u_n)\frac{u_n}{\|u_n\|}\,dx,\\
D_{K,n}=\int_{\{x\in\Omega:u_n(x)<-K,v(x)<0\}}F(u_n)\frac{u_n}{\|u_n\|}\,dx, \\
E_{K,n}=\int_{\{x\in\Omega:u_n(x)<-K,v(x)\ge0\}}F(u_n)\frac{u_n}{\|u_n\|}\,dx.
\end{gather*}
Before estimating these integrals we claim that for any $K>0$ the following
assertions are true, since that $\|u_n\|\to +\infty$ and $u_n/\|u_n\|\to v$
in $W_0^{1,p}(\Omega)$ as $n\to\infty$.
%%
\begin{gather}
\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)\le K,v(x)>0\}}v_n\,dx=0,
\label{lim-1} \\
\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)>K,v(x)\le0\}}v_n\,dx=0,
\label{lim-2}\\
\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)\ge -K,v(x)<0\}}v_n\,dx=0,
\label{lim-3}\\
\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)< -K,v(x)\ge0\}}v_n\,dx=0.
\label{lim-4}
\end{gather}
%%
In fact, for the first equality (\ref{lim-1}), we have
\begin{align*}
&\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)\le K,v(x)>0\}}v_n\,dx\\
&= \lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)<-K,v(x)>0\}}v_n\,dx
 +\lim_{n\to \infty}\int_{\{x\in\Omega:-K\le u_n(x)\le K,v(x)>0\}}v_n\,dx
\\
&= \lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)<-K,v(x)>0\}}v_n\,dx
\le 0.
\end{align*}
Moreover, since $v_n\to  v$ in $L^p(\Omega)$, it follows that
\[
\int_{\{x\in\Omega:u_n(x)<-K,v(x)>0\}}|v_n-v|\,dx
\le |\Omega|^{1-1/p}\|v_n-v\|_{L^p}\to  0,
\quad\hbox{as }n\to \infty,
\]
which implies
$$
0\ge\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)<-K,v(x)>0\}}v_n\,dx
=\lim_{n\to \infty}\int_{\{x\in\Omega:u_n(x)<-K,v(x)>0\}}v\,dx
\ge0,
$$
and so proves the limit equality (\ref{lim-1}).
For the other three equalities (\ref{lim-2})--(\ref{lim-4}), the proofs are
similar and we omit the details.
Furthermore, have
\begin{gather*}
|A_{K,n}|\le\frac{K c(K)|\Omega|}{\|u_n\|}\to 0,
 \\
\begin{aligned}
B_{K,n}&\ge  c_\varepsilon\Big(\int_{\{x\in\Omega:v(x)>0\}}v_n\,dx-
\int_{\{x\in\Omega:u_n(x)\le K,v(x)>0\}}v_n\,dx\Big)\\
&\to  c_\varepsilon\int_{\{x\in\Omega:v(x)>0\}}v\,dx,
\end{aligned}\\
C_{K,n}\ge  c_\varepsilon\int_{\{x\in\Omega:u_n(x)>K,v(x)\le0\}}v_n\,dx\to  0,
\\
\begin{aligned}
D_{K,n}&\ge  d_\varepsilon\Big(\int_{\{x\in\Omega:v(x)<0\}}v_n\,dx-
\int_{\{x\in\Omega:u_n(x)\ge -K,v(x)<0\}}v_n\,dx\Big) \\
&\to  d_\varepsilon
\int_{\{x\in\Omega:v(x)<0\}}v\,dx,
\end{aligned}\\
E_{K,n}\ge  d_\varepsilon
\int_{\{x\in\Omega:u_n(x)< -K,v(x)\ge0\}}v_n\,dx\to  0.
\end{gather*}
Recalling (\ref{3.14}), for $\varepsilon>0$, we obtain
%%
\begin{align*}
&\liminf\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx \\
&= \liminf(A_{K,n}+B_{K,n}+C_{K,n}+D_{K,n}+E_{K,n})\\
&\ge c_\varepsilon\int_{\{x\in\Omega:v(x)>0\}}v(x)\,dx
+d_\varepsilon\int_{\{x\in\Omega:v(x)<0\}}v(x)\,dx.
\end{align*}
%%
By the definition of $c_\varepsilon$ and $d_\varepsilon$
together with (\ref{3.11}) and the above inequality, we conclude that
$$
(p-1)\int_\Omega h(x)v(x)\,dx\ge\underline{F(+\infty)}\int_\Omega v^+(x)\,dx
+\overline{F(-\infty)}\int_\Omega v^-(x)\,dx,
$$
clearly which contradicts \eqref{1-4}, and so we complete the proof of
the boundedness of $\{u_n\}$.

Since $\{u_n\}$ is bounded in $W^{1,p}_0(\Omega)$, then there
exists $u\in W^{1,p}_0(\Omega)$, such that (passing to subsequences)
%%
\begin{equation}
\label{aa}
u_n\rightharpoonup u\quad\hbox{in }W^{1,p}_0(\Omega),\quad
u_n\to  u\quad\hbox{in }L^p(\Omega).
\end{equation}
Taking (\ref{3.2}) and \eqref{1-3} into account, it follows that
%%
\begin{align*}
0&=\lim\langle J'_\lambda(u_n),u_n-u\rangle\\
&=\lim\int_\Omega|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n-\nabla u)\,dx-\lambda
\int_\Omega|u_n|^{p-2}u_n(u_n-u)\,dx\\
&\quad -\int_\Omega g(u_n)(u_n-u)\,dx+\int_\Omega h(u_n-u)\,dx.
\end{align*}
Recalling \eqref{1-3} and combining with the continuity of $g(t)$,
we have that for any $\varepsilon>0$, there exists $M>0$, such that
$|g(u_n)|\le M+\varepsilon|u_n|^{p-1}$, which together with (\ref{aa})
yield that the last three terms goes to zero, and
$$
\lim\int_{\Omega}|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n-\nabla u)\,dx=0.
$$
Similar to (\ref{3.7}), we obtain $\|u_n\|\to \|u\|$.
The uniform convexity of $W^{1,p}_0(\Omega)$ then yields $u_n\to  u$
in $W^{1,p}_0(\Omega)$, which complete the proof.
\end{proof}


Next, we prove the main theorem. As in \cite{1}, we divide it into three
lemmas for different cases separately:
$$
\lambda<\lambda_1, \quad \lambda_k<\lambda<\lambda_{k+1}, \quad
\lambda=\lambda_k.
$$

\begin{lemma} \label{Lem-2}
Assume \eqref{1-3} holds, and $\lambda<\lambda_1$. Then \eqref{1-1}
admits at least one weak solution.
\end{lemma}

\begin{proof}
By the definition of $J_\lambda(u)$ and the assumption on $g(t)$, for any
$\varepsilon>0$ we have
%%
\begin{align*}
J_\lambda(u)&=  \frac1p\int_\Omega|\nabla u|^p\,dx-\frac\lambda p\int_\Omega|u|^p\,dx
-\int_\Omega G(u)\,dx+\int_\Omega h(x)u(x)\,dx
\\
&\ge  \frac{\lambda_1-\lambda} p\int_\Omega|u|^p\,dx
-C\int_\Omega|u|\,dx-\frac{\varepsilon}{p}\int_\Omega|u|^p\,dx-\int_\Omega |h(x)u(x)|\,dx
\\
&\ge \frac{\lambda_1-\lambda-\varepsilon} p\|u\|^p_{L^p(\Omega)}-
C\|u\|_{L^1(\Omega)}-\|h\|_{L^{p'}}\|u\|_{L^p(\Omega)},
\end{align*}
which implies that the functional $J_\lambda$ is bounded from below on
$W_0^{1,p}(\Omega)$. Moreover, from Lemma \ref{Lem-1}, we have
$J_\lambda$ satisfies the Palais-Smale condition. Hence
$J_\lambda$ attains its global minimum on $W_0^{1,p}(\Omega)$.
\end{proof}

\begin{lemma} \label{Lem-3}
Assume \eqref{1-3}, \eqref{1-4} (or \eqref{1-5}) hold, and there exists
$k\in \mathbb N$ such that $\lambda_k<\lambda<\lambda_{k+1}$. Then \eqref{1-1}
admits at least one weak solution.
\end{lemma}

\begin{proof}
Let $m\in (\lambda_k, \lambda)$, and let $\mathcal A\in \mathcal F_k$,
such that
$\sup\limits_{u\in \mathcal A} R(u)\le m$.
Then for all $u\in\mathcal A$, $t>0$ and all $\varepsilon>0$,
 by \eqref{1-3} there exists $c>0$, such that
%%
\begin{align*}
J_\lambda(tu) &=  \frac 1p t^p\Big(\int_\Omega|\nabla u|^p\,dx-
\lambda\int_\Omega|u|^p\,dx\Big)-\int_\Omega G(tu)\,dx
 +t\int_\Omega h(x)u(x)\,dx\\
&\le  \frac 1p t^p(m-\lambda)\|u\|^p_{L^p(\Omega)}+ct\|u\|_{L^1(\Omega)}+
\frac\varepsilon p t^p\|u\|^p_{L^p(\Omega)}
 +t\|h\|_{L^{p'}(\Omega)}\|u\|_{L^p(\Omega)}\\
&= \frac 1p t^p(m-\lambda+\varepsilon)\|u\|^p_{L^p(\Omega)}
+t(c\|u\|_{L^1(\Omega)}+ \|h\|_{L^{p'}(\Omega)}\|u\|_{L^p(\Omega)}).
\end{align*}
Clearly,
%%
\begin{equation} \label{3-1}
\lim_{t\to+\infty}J_\lambda(tu)= -\infty \quad \text{uniformly for any }
\ u\in \mathcal A.
\end{equation}
%%
Now let
$$
\varepsilon_{k+1}:=\{u\in W_0^{1,p}(\Omega); \int_{\Omega}|\nabla u|^p\,dx\ge
\lambda_{k+1}\int_\Omega|u|^p\,dx\}.
$$
By noting that for all $u\in \varepsilon_{k+1}$, and all $\varepsilon>0$,
there exists $c>0$, such that
$$
J_\lambda(u)\ge \frac 1p(\lambda_{k+1}-\lambda-\varepsilon)\|u\|^p_{L^p(\Omega)}
-c\|u\|_{L^1(\Omega)}-\|h\|_{L^{p'}(\Omega)}\|u\|_{L^p(\Omega)}.
$$
Hence $J_\lambda(u)$ is bounded from below in $\varepsilon_{k+1}$.
Let
%%
\begin{equation}\label{3-2}
\alpha=\inf_{u\in \varepsilon_{k+1}}J_\lambda(u).
\end{equation}
 From (\ref{3-1}) and (\ref{3-2}), we see that there exists $T>0$ such that
$$
\gamma:=\max\{J_\lambda(tu); \ u\in\mathcal A,\ t\ge T\}< \alpha.
$$
Define
\begin{gather*}
T\mathcal A:=\{tu\in W_0^{1,p}(\Omega); \ u\in\mathcal A, \ t\ge T\}, \\
\Gamma:=\{h\in C^0(B_k,W_0^{1,p}(\Omega)); \ h|_{\mathcal{S}_{k-1}}
\to T\mathcal A \text { is an odd map}\},
\end{gather*}
where $B_k$ is a unit ball centered at the origin in $\mathbb{R}^k$.
Then we see that $\Gamma$ is nonempty.
In fact, recalling the definition of $\mathcal F_k$, we see that
there exists a continuous
odd surjection
$h: \mathcal{S}_{k-1}\to \mathcal A$. Define
\begin{gather*}
\overline h: B_k\to W_0^{1,p}(\Omega), \\
\overline h(tx)=tTh(x) \quad \text{for } x\in\mathcal{S}_{k-1},\; t\in [0,1].
\end{gather*}
Obviously, $\overline h\in\Gamma$.
Furthermore, if $h\in\Gamma$, then
%%
\begin{equation} \label{3-3}
h(B_k)\cap\varepsilon_{k+1}\not=\phi.
\end{equation}
%%
In fact, if $0\in h(B_k)$, then (\ref{3-3}) holds clearly.
Otherwise, considering the mapping
$\widetilde h: \mathcal{S}_{k}\to \mathcal{S},$
\[
\widetilde h(x_1,\dots, x_{k+1})=
\begin{cases}
\pi\cdot h(x_1,\dots, x_{k}), & x_{k+1}\ge 0, \\
-\pi\cdot h(-x_1,\dots, -x_{k}),  & x_{k+1}< 0,
\end{cases}
\]
where $\pi$ represents radial projection onto $\mathcal{S}$ in
$W_0^{1,p}(\Omega)\backslash\{0\}$,
clearly, we have $\widetilde h(\mathcal{S}_k)\in \mathcal F_{k+1}$.
 From the definition of $\lambda_{k+1}$, we see that
$$
\sup_{u\in\widetilde h(\mathcal{S}_k)} R(u)\ge\lambda_{k+1},
$$
which implies that there exists
$u=\pi\cdot h(x)\in\widetilde h(\mathcal{S}_k)$
such that $R(u)\ge \lambda_{k+1}$.
That is $u=\pi\cdot h(x)\in \varepsilon_{k+1}$,
which also implies that $h(\bar x)\in \varepsilon_{k+1}$, where
$\bar x= x/\|x\|$. Thus
$h(B_k)\cap\varepsilon_{k+1}\not=\phi$.

Moreover, recalling the Deformation Lemma, we see that
$$
C=\inf_{h\in\Gamma}\sup_{x\in B_k}J_\lambda (h(x))
$$
is a critical value of $J_\lambda$. In fact, we assume by contradiction that
$C$ is a regular value of $J_\lambda$, from
$h(B_k)\cap \varepsilon_{k+1}\not=\phi$, it is easy to see that
$C\ge \alpha>\gamma$. Let $\overline\varepsilon$ be an arbitrary given
constant in $(0,C-\gamma)$.
By the definition of $C$, for any $\varepsilon\in (0,\overline\varepsilon)$,
there exists a corresponding $h\in\Gamma$, such that
$$
\sup_{x\in B_k}J_\lambda (h(x))<C+\varepsilon.
$$
Then by the Deformation Lemma, there exists $\varepsilon$ and a
corresponding
$\varphi: W_0^{1,p}(\Omega)\times [0,1]\to W_0^{1,p}(\Omega)$ such that
$$
J_\lambda(\varphi(h, 1))\le C-\varepsilon.
$$
For any $x\in\mathcal{S}_{k-1}$, $h(x)\in T\mathcal A$,
$$
J_\lambda(h(x))<\gamma<C-\overline\varepsilon.
$$
Hence, $\varphi(h,1)=h\in \Gamma$, which contradicts the definition of
$C$.
\end{proof}

\begin{lemma} \label{lem-4}
Let us assume \eqref{1-3}, \eqref{1-4} or (\eqref{1-5}), and there exists
$k\in \mathbb N$ such that $\lambda=\lambda_k$. Then \eqref{1-1} admits
at least one weak solution.
\end{lemma}

\begin{proof}
We split the proof into several steps, in the first step, we show the case of
\eqref{1-4}, then the second step is devoted to the case of \eqref{1-5}.

\noindent \textbf{Step 1.} Assume \eqref{1-4}. Take sequence $\{\mu_n\}$ with
$\lambda_k<\mu_n<\lambda_{k+1}$ and $\mu_n\searrow\lambda_k$. By means of Lemma
\ref{Lem-3}, there exists a sequence $\{u_n\}$ of critical points associated with
the functional $\{J_{\mu_n}\}$ such that
$$
C_n=J_{\mu_n}(u_n)\ge \alpha_n: =\inf\{J_{\mu_n}(u): u\in\varepsilon_{k+1}\}.
$$
For all $u\in \varepsilon_{k+1}$,
%%
\begin{align*}
J_{\mu_n}(u)=\frac1p\int_\Omega|\nabla u|^p\,dx-\frac{\mu_n} p\int_\Omega|u|^p\,dx
-\int_\Omega G(u)\,dx+\int_\Omega h(x)u(x)\,dx
\\
\ge \frac1p(\lambda_{k+1}-\mu_n-\varepsilon)\|u\|_{L^p}^p(\Omega)-
C\|u\|_{L^1(\Omega)}-\|h\|_{L^{p'}}\|u\|_{L^p},
\end{align*}
which implies that $C_n$ is bounded from below uniformly.

In the following, we pay our attention to the boundedness of the corresponding
sequence of critical points $\{u_n\}$.
Suppose to the contrary, there exists a subsequence of $\{u_n\}$, for simplify,
we might as well assume to be itself, such that $\|u_n\|\to \infty$. Similar to
Lemma \ref{Lem-1}, we can show that there exists  $v\in \ker (-\Delta_p-\lambda_k)
\backslash\{0\}$, such that (up to subsequence) $\frac{u_n}{\|u_n\|}\to v$.
Since $C_n$ is bounded from below, then we have
%%
\begin{align*}
0&\le  \liminf\frac {pC_n}{\|u_n\|}\le\limsup\frac {pC_n}{\|u_n\|}
\\
&=  \limsup\frac{pJ_{\mu_n}(u_n)-\langle J_{\mu_n}'(u_n), u_n\rangle}{\|u_n\|}
\\
&=  \limsup\Big(-\frac{p\int_\Omega G(u_n)\,dx-\int_{\Omega}g(u_n)u_n\,dx}{\|u_n\|}
+(p-1)\int_{\Omega}hv_n\,dx\Big)
\\
&=  -\liminf\Big(\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx\Big)+(p-1)
\int_{\Omega}hv\,dx.
\end{align*}
%%
Similar to Lemma \ref{Lem-1}, we obtain
$$
\underline {F(+\infty)}\int_{\Omega}v^+(x)\,dx
+\overline {F(-\infty)}\int_{\Omega}v^-(x)\,dx\le (p-1)\int_{\Omega}h(x)v(x)\,dx,
$$
which contradicts to the assumption \eqref{1-4}, that is $\{u_n\}$
is bounded in $W_0^{1, p}(\Omega)$. Thus, there exists
$u\in W_0^{1, p}(\Omega)$, such that (passing to subsequence)
$$
u_n\rightharpoonup u \ \text{in } W_0^{1,p}(\Omega),
\quad u_n\to u  \quad \text{in } L^p(\Omega).
$$
Therefore,
%%
\begin{align*}
0&=\lim_{n\to\infty}\langle J_{\mu_n}'(u_n), u_n-u\rangle\\
&= \lim_{n\to\infty}\int_\Omega|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n
  -\nabla u)\,dx
  -\mu_n\int_\Omega|u_n|^{p-2}u_n(u_n-u) \,dx\\
&\quad -\int_{\Omega}g(u_n)(u_n-u)\,dx +\int_\Omega h(u_n-u)\,dx\\
&= \lim_{n\to\infty} \int_\Omega|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n
  -\nabla u)\,dx.
\end{align*}
%%
Recalling H\"older's inequality, we conclude that
%%
\begin{align*}
0&\leftarrow \int_\Omega|\nabla u_n|^{p-2}\nabla u_n(\nabla u_n-\nabla u)\,dx
 -\int_\Omega|\nabla u|^{p-2}\nabla u(\nabla u_n-\nabla u)\,dx
\\
&=  \int_\Omega|\nabla u_n|^p \,dx-\int_\Omega|\nabla u_n|^{p-2}\nabla u_n\nabla u\,dx
  -\int_\Omega|\nabla u|^{p-2}\nabla u\nabla u_n\,dx
+\int_\Omega|\nabla u|^p \,dx
\\
&\ge \|u_n\|^p-\|u_n\|^{p-1}\|u\|-\|u\|^{p-1}\|u_n\|+\|u\|^p
\\
&= (\|u_n\|^{p-1}-\|u\|^{p-1})(\|u_n\|-\|u\|)\ge 0,
\end{align*}
%%
which implies that $\|u_n\|\to\|u\|$.
The uniform convexity of $W_0^{1,p}(\Omega)$ yields
$$
u_n\to u \quad \text{in }\ W_0^{1,p}(\Omega).
$$
Considering the sequence $\{J_{\mu_n}(u_n)\}$ (passing to a subsequence if necessary),
letting $n\to\infty$,
and combining with the Lebesgue dominated convergence theorem,
we finally arrive at
$$
J_{\mu_n}(u_n)\to J_{\lambda_k}(u)=C \text{ and } \ J_{\lambda_k}'(u)=0,
$$
which implies that
$u$ is a critical point of $J_{\lambda_k}$.

\noindent\textbf{Step 2.}
Next, we transfer our attention to the case of \eqref{1-5}.
First of all, we consider the case of $k=1$.
Take sequence $\{\mu_n\}$ with  $0<\mu_n<\lambda_1$ and
$\mu_n\nearrow\lambda_1$.
We can find a sequence $\{u_n\}$ of critical points associated with the
functional
$\{J_{\mu_n}\}$ such that $C_n=J_{\mu_n}(u_n)$ is decreasing.
Now we are going to show $\{u_n\}$ is bounded. Suppose, by contradiction,
$\|u_n\|\to \infty$, then there exists
$v\in \ker (-\Delta_p-\lambda_1)\backslash\{0\}$ such that (up to a subsequence)
$u_n/\|u_n\| \to v$, and
%%
\begin{align*}
0&\ge \limsup\frac {pC_n}{\|u_n\|}\\
&\ge\liminf\frac {pC_n}{\|u_n\|}\\
&=  \liminf\frac{pJ_{\mu_n}(u_n)-\langle J_{\mu_n}'(u_n), u_n\rangle}{\|u_n\|}
\\
&=  \liminf\Big(-\frac{p\int_\Omega G(u_n)\,dx-\int_{\Omega}g(u_n)u_n\,dx}{\|u_n\|}
+(p-1)\int_{\Omega}h\frac{u_n}{\|u_n\|}\,dx\Big)
\\
&=  -\limsup\Big(\frac{p\int_\Omega G(u_n)\,dx-\int_{\Omega}g(u_n)u_n\,dx}{\|u_n\|}
\Big)+(p-1)\int_{\Omega}hv\,dx
\\
&=  -\limsup\Big(\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx\Big)+(p-1)
\int_{\Omega}hv\,dx>0,
\end{align*}
%%
which is a contradiction.  The following argument is completely
parallel to Step 1, so we omit it.

In the following, we focus on the case of $k>1$. Let
$\{\mu_n\}$ be a sequence in  $(\lambda_{k-1},\lambda_k)$ with $\mu_n\nearrow\lambda_k$.
We can find a sequence $\{u_n\}$ of critical points associated with the functional
$\{J_{\mu_n}\}$ such that $C_n=J_{\mu_n}(u_n)$ is decreasing.
Then we obtain that $\{u_n\}$ is bounded. Suppose, by contradiction,
$\|u_n\|\to \infty$, then there exists
$v\in \ker (-\Delta_p-\lambda_k)\backslash\{0\}$ such that (up to subsequence)
$\frac{u_n}{\|u_n\|}\to v$, and
%%
\begin{align*}
0&\ge  \limsup\frac {pC_n}{\|u_n\|}\\
&\ge\liminf\frac {pC_n}{\|u_n\|}
\\
&=  \liminf\frac{pJ_{\mu_n}(u_n)-\langle J_{\mu_n}'(u_n), u_n\rangle}{\|u_n\|}
\\
&=  \liminf\Big(-\frac{p\int_\Omega G(u_n)\,dx-\int_{\Omega}g(u_n)u_n\,dx}{\|u_n\|}
+(p-1)\int_{\Omega}h\frac{u_n}{\|u_n\|}\,dx\Big)
\\
&=  -\limsup\Big(\frac{p\int_\Omega G(u_n)\,dx-\int_{\Omega}g(u_n)u_n\,dx}{\|u_n\|}
\Big)+(p-1)\int_{\Omega}hv\,dx
\\
&=  -\limsup\Big(\int_\Omega F(u_n)\frac{u_n}{\|u_n\|}\,dx\Big)+(p-1)
\int_{\Omega}hv\,dx>0,
\end{align*}
%%
which is a contradiction. The remaining argument is quite simple, similar to the
above discussion, and so we omit it here.
\end{proof}

\begin{proof}[Proof of Theorem \ref{1.1}]
Combining Lemma \ref{Lem-2} -- Lemma \ref{lem-4},
Theorem \ref{1.1} holds clearly. The proof is complete.
\end{proof}

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\end{document}